mediate Course 
Me nical Drawing 






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LIBRARY OF CONGRESS. 



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UNITED STATES OF AMERICA. 



INTERMEDIATE COURSE 



IN 



MECHANICAL DRAWING, 



WILLIAM H. XHORNE, 

Director of the Drawing School of the Franklin Institute of Philadelphia, 
Member of the American Society of Mechanical Engineers. 



SECOND EDITION. 

/ 

PUBLISHED BY 

WILLIAMS, BROWN & EARLE, 
N. E. Cor. Chestnut and Tenth Sts., Philadelphia. 

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4V 



Entered according to Act of Congress, in the year 1 891, by 

WILLIAM H. THORNE, 
In the Office of the Librarian of Congress, at Washington. 



My intention, in arranging this Intermediate Course in Mechanical Draw- 
ing, has been to present the subject of Orthographic Projection in a concise, 
logical and comprehensive manner, keeping theory always in view, but, at 
the same time, doing everything accordiug to methods which practical ex- 
perience has proven to be the easiest, most accurate, useful and readily inter- 
preted. 

The endeavor has been to avoid giving, on the one hand, a mass of defini- 
tions, rules, theorems and analyses, which are so readily forgotten, and on the 
other hand, minute directions for every line, which would make the study a 
mere copying process ; but to encourage and induce the interest, close attention 
and thought of the student, and thus bring about a thorough comprehension of 
the theory and principles and a correct training for the practice of Mechanical Draw- 
ing, so that all the apparently new conditions, which are constantly arising, can be 
analyzed, not by reference to text-book rules of doubtful applicability, but by the 
exercise of the individual reasoning powers. 

WM. H. THORNE. 

Gowen Ave., Mount Airy, 

Philadelphia, 1889. 



INTRODUCTION. 

As has been explained in the Junior Course, the purpose of Mechanical 
Drawing is to give such an illustration of a required object as to enable it to be 
accurately and definitely built from the drawing alone. To accomplish this 
purpose, the drawing must contain a sufficient number of views of the object 
to show the true size of every feature necessary for the information of the 
person who is to build it. It is not sufficient that the lines and surfaces be 
fully determined geometrically, but their dimensions and relative positions must 
also be shown in the way to be most easily understood and to require the least 
effort of thought in the interpretation of the drawing. The thinking must be done 
by the draughtsman, at least as to the form, size and purpose of the structure, 
and the drawing should embody that thought in every particular. Mechanical 
Drawing is an embodiment of thought, and, largely, of original thought, com- 
bined with technical knowledge and manual training. Hence, anything like 
copying should be studiously avoided when the theory and principles are being 
studied, and the mind should be trained from the start to work from a concep- 
tion, an idea, and to put that idea on paper so as to be understood. The 
imagination, or what might be called the scientific imagination, must be con- 
tinually on the alert, because the hand is constantly called upon to do something 

5 



6 INTERMEDIATE COURSE. 

that it has never done before, the different conditions and combinations which 
are constantly arising being infinite in their variety. Drawing from models 
should also be avoided. They may be used at the beginning for the purpose 
of giving a mental conception of the object before the student's mind has been 
trained to form a clear idea of it from a verbal description, or before his inven- 
tive faculties have been sufficiently developed to enable him to originate one in 
his own mind ; but he should soon be made to realize the proper sequence of 
the art, which is : first, the conception, either original or derived from another ; 
and, second, the representation of that conception, by means of a drawing, in 
such a clear, explicit and accurate manner as to enable an artisan to produce it 
in the concrete. The habit of drawing from models is the reverse of this, and 
does not tend to properly develop the imagination or train the mind for original 
work or produce good methods. Copying from other drawings must only be 
done for the purpose of gaining experience of the principles involved and the 
style of execution ; but in every case another drawing should be made, with the 
proportions and conditions so changed as to compel an exercise of original 
thought. All drawings should be inked, and inked properly, imaginary lines 
being distinguished from actual lines. After spending time, work and thought 
on a drawing, it is simply folly to leave it in a condition which is unintelligible 
to any one but the author, and which will soon become so to him. The addi- 
tional time required to ink, finish and dimension the drawing is amply repaid 



MECHANICAL DBA WING. ' 7 

by the clearness, permanence and usefulness of the result. Never make a draw- 
ing, not even a purely theoretical one, without using definite measurements, and 
always put on the essential dimensions. This habit should be cultivated from 
the start, as it tends to accuracy of thought and workmanship, and is a training 
for the proper selection of the parts requiring dimensions and of the best places 
to mark them. 

The dimension lines and all imaginary lines, or those which do not repre- 
sent parts of the object itself, but serve as bases or lines of reference, or which 
are necessarily used in the construction of the drawing and the preservation of 
which would be desirable, should be distinguished from the black lines of the 
object in such manner as not to interfere with the clearness of the latter. In 
other words, the object should stand out boldly and clearly to permit a ready 
general comprehension of it, while the detailed information concerning it should 
be kept somewhat in the background. The importance of this is not so appar- 
ent in the drawing of the simple geometrical solids in the following problems, 
but it is very great in all work of any intricacy. 

All the devices which are used by intelligent and experienced draughtsmen 
to add to the usefulness, clearness and beauty of their work, should be employed 
by the student, in order that their use may become, in a manner, intuitive, and thus 
leave the mind free for concentration on the construction. 



INTERMEDIATE COUKSE. 



PROJECTIONS. 

The Junior Course has shown that Mechanical Drawings are made upon 
the theory that the imaginary object is surrounded by imaginary planes perpen- 
dicular to each other, and that lines perpendicular to these planes are projected 
from each point of the object to each of the planes, and that the points where 
the projecting lines pierce the planes are the projections on these planes of the 
points of the object, and that these planes are then supposed to be revolved 
upon their intersections to bring them all into one plane — that of the paper. 
The projections on each of these planes thus form a distinct view from one 
direction, and give the appearance which the object would present if viewed 
from an infinite distance in a direction perpendicular to the plane, the plane 
being between the object and the point of sight. The revolution of these planes 
into the plane of the paper brings the several views into the most convenient 
and sensible positions in relation to each other. It brings the Plan or Top View 
above, the Right Side View to the right, the Left Side View to the left, and any 
oblique view immediately adjacent to the part it represents. These are the 
positions which experience in the making of intricate drawings has proven to 
be the clearest and most manageable, and to be preferable to the opposite 
system, which imagines the object to be between the plane of projection and 



MECHANICAL DRAWING. 9 

the point of sight, with the result of locating the views in the opposites of these 
positions ; that is, a view of the right side would be at the left, a view of the 
top would be underneath, and so on. Advocates of the latter system rarely 
adhere rigidly to it, and, in drawing oblique views, almost invariably violate it, 
and this fact is one of the strongest reasons against using it. Apart from its 
inconvenience, there is no vital objection to it provided that it is rigidly adhered 
to, because then the location of a view will immediately and positively indicate 
which side of the object it represents ; but if the two systems are both used in 
the same drawing, doubt and possibly errors will result. To those acquainted with 
Descriptive Geometry, the method here advocated and invariably employed is the use 
of the third an^le and not the first. 

The actual indication of the planes of projection, by showing their intersections 
or the axes about which they are supposed to have been revolved into the plane 
of the paper, has already been abandoned in the Junior Course, and will not be 
used at all in this Intermediate Course, as it is desirable that the mind should be 
trained to recognize the relation of one view to another without the intervention of 
these axes of projection. 

The number of possible views of the same object is infinite, and only such 
as will show in the simplest manner the essential dimensions and form should 
be selected. It frequently occurs, however, that one detail or unit of a structure 
is oblique to the main body, necessitating oblique views. Figs. 59 and 60 are 



10 INTERMEDIATE COURSE. 

given as examples of such cases, and also for the purpose of showing the variety of 
different views which can be made and their proper relation to each other. 



TECHNICALITIES. 

Each Plate represents a sheet of drawing paper 16 by 21 inches, with 
margin lines 15 by 20. The Figures are one-fourth size, but are to be drawn 
full size. 

All the lines representing the Object are to be inked black, those repre- 
senting visible parts being full lines, and those representing hidden parts being com- 
posed of short dots, as shown by the full lines and short-dotted lines in the Plates. 
Shade Lines are to be used in all cases, and to be located on the theory that the 
light falls upon the drawing from the upper left hand at an angle of 45°, and pro- 
duces the same effect on all the different views, namely : of making the lower and 
right-hand edges shaded. 

Circles and curves are to be inked first, and each shaded immediately, then all 
the straight fine and dotted lines, and finally the heavy shade lines. 

After completing the black lines, ink all the Centre Lines or axes of 
symmetry and any important lines of reference, blue : lastly, ink all the dimen- 
sion lines and any construction lines used in obtaining the lines of the object, 



MECHANICAL DRAWING. 11 

the preservation of which is desirable, red. The dimensions and the arrow-heads or 
points at the extremities of the dimension lines should be black. 

In the accompanying Plates, the Figures being printed entirely in black, the 
blue centre lines are represented by long dots, the red construction lines by long- 
and-short dots, and the red dimension lines can not be mistaken. In making a 
drawing, however, the blue centre lines and red construction lines must not be dotted, 
but made full lines. 

For directions as to handling instruments and materials, refer to pages 5, 6, 8, 
30, 31 and 32 of the Junior Course. 

Adopt some neat style of lettering that can be easily and quickly executed, 
and form the habit of always putting a signature and date as well as a title upon 
all drawings, in order to insure their identification ; but never make these conspicuous, 
because the primary object is to show the structure, to which everything should be 
made subservient. 

Neat execution and artistic effect are desirable quantities in a drawing, but 
correctness, clearness, and an emphatic, unquestionable expression of precisely 
what is meant, are still more important. It is the beauty of the conception and 
the exactness with which the result can be produced in the concrete, which con- 
stitutes the beauty of a mechanical drawing, and, therefore, any attempt at scenic 
effect is not only in bad taste, but often detracts from its usefulness. If, however, a 
picture of the structure is required for the information of those who are unable to 



12 INTERMEDIATE COURSE. 

understand a working drawing, or to conceive of what it represents, then artistic effect 
becomes the principal object, 



OBLIQUE VIEWS. 

PLATE 9. 

Fig. 59. To make a mechanical drawing of a structure which is oblique to the 
vertical planes of projection. 

This condition would, of course, occur only when the oblique structure was but a 
part of a complicated whole, the more important features of which were parallel to 
the planes of projection. 

Let the oblique structure have a horizontal rectangular base 3J r/ long and If" 
wide, the longitudinal centre line of the base making an angle of 30° with the front 
vertical plane. Let the two sides and the two ends of the structure incline at 60° 
with the base, and let the base, sides and ends be \" thick. 

In the first place, determine a convenient position on the drawing paper for 
the Plan or Top View, and locate the centre p of the base, through which draw 
the vertical line ah for the trace of a vertical plane perpendicular to the front 



MECHANICAL DRAWING. 13 

elevation, and the horizontal line cd for the trace of a vertical plane parallel to the 
front elevation. These two lines then represent two vertical planes perpendicular to 
each other and to the planes of projection, and passing through the centre of the 
structure, and will serve as bases from which to locate the points. 

Through the central point p, at the given angle, 30° with cd, draw gh for 
the longitudinal centre line of the base, and perpendicular to gh draw ef for the 
transverse centre line. On these centre lines, about the central point p, draw the base 
3J" by If", as given. We now have the traces of four vertical planes intersecting 
at the centre or axis of the structure, two of which are parallel to the planes of pro- 
jection and two normal to the sides of the structure, and we have the outline of the 
base on a horizontal plane. Before we can obtain the front and side elevations, 
which will be oblique views, we must first draw the projections upon Auxiliary 
Planes parallel to the central planes of the structure, because only upon such planes 
does it appear in its true form and size. Hence, project an end view, B, of the 
base by drawing a line equal to its width parallel to ef, and from the extremities of 
this line draw the inclined sides, and, parallel with these sides and the base and \" 
from them, draw the interior lines for their thickness. 

It is a fundamental principle of mechanical drawing that each view should be 
made to give all the useful information that it can in the clearest manner. 
In the end view which we have just drawn, the additional information beyond 
that given by the plan, is the inclination of the sides and the thickness of the 



14 INTERMEDIATE COURSE. 

sides and base. It is evident that this thickness can be shown more clearly by 
representing the structure as cut through to expose it, than by indicating it by 
dotted lines. The representation of such a cut is called a Section, and is made by 
equidistant fine black lines at 45° with the principal side. The distance between 
the lines varies from -fa" to \" y according to the scale of the drawing and the size 
of the object, and is determined by judgment and taste. The Section is supposed to 
be on the central plane unless otherwise indicated, and is always on a "plane parallel 
to the plane of projection of the view showing it. In the present instance it is a 
Vertical Section on the plane ef 

From this end view, _B, and the plan of the base, A, project an Auxiliary 
Front View, C } by drawing the trace, e'f, of the vertical plane, ef and projecting 
the end view of the base and intersections of the sides across it, laying off half 
the length of the base on each side of ef , and from the extremities of the base 
drawing lines of the given inclination for the ends. The interior can be dotted or 
the view shown in Section, whichever appears most desirable. Frequently the details 
of a structure are such that, to insure clearness, it is necessary to make both an 
external view and a section on the same plane of projection, the views C and D show- 
ing their arrangement in the present instance, although in this Figure there is no 
necessity whatever for both. 

Now complete the plan," A, by projecting to it the external and internal 
intersections of the sides of the end view, and on these projecting lines laying off 



MECHANICAL DRAWING. 15 

the lengths of the intersections as obtained from the Auxiliary Front View, C or D 
(remembering that any point is always at the same perpendicular distance from ef that 
it is from e'f\ and drawing the diagonal lines for the intersections of the ends with 
the sides. 

We now have all the data for the completion of the front and side views E and 
F f originally required, for which purpose draw a line perpendicular to ab for the 
horizontal base of the structure, and from this base lay off on ab the external and 
internal heights taken from view 0, and draw indefinite lines for these heights. Con- 
tinue these lines for the heights in the side view F. Project all the points down from 
the plan A, to the front view E. In the side view F, draw the line c'd' for the 
trace of the vertical plane cd, and locate all points at the same perpendicular distance 
from c'd' as they are from cd. 

In case a vertical section on plane ab were desirable, an additional Side 
View could be drawn in the position shown at G, with c n d" as the trace of the 
plane cd. 

This Figure shows how a number of views can be projected one from the other 
so that, no matter how complicated or oblique a structure may be, all of its parts may 
be clearly and definitely shown in true proportions and proper relation, the general 
principle being that all the planes of projection in a series must be perpendicular to each 
other, and that a new series of planes can be started by a plane perpendicular to either 
of (he planes of the first series and, oblique to all the rest 



16 INTERMEDIATE COURSE. 

A knowledge of the use of Auxiliary Views is very important, as, by their means, 
oblique details of structures can be readily designed and drawn so as to maintain their 
proper relation to the other views. In order to make this relation more clear, Fig. I 
shows a perspective drawing of the same structure as that just completed by Fig. 59, 
surrounded by all the planes of projection which were used in the latter, namely, the 
one horizontal and the four vertical planes comprising the top, front and side views 
and the two auxiliary views. Fig. m shows these vertical planes revolved up into the 
horizontal plane of the paper, and is a perspective drawing of a mechanical drawing 
of the imaginary solid shown below them. It is evident that an infinite number of 
vertical auxiliary views can be used, also that auxiliary views which are not vertical 
can be readily drawn perpendicular to any plane which is already established, as for 
instance, perpendicular to the end auxiliary view and parallel to the inclined front of 
the structure, for the purpose of showing the true shape and size of this front or for 
designing any ornamentation for it. 

After the imagination is trained to picture these planes, no trouble need be 
anticipated in arranging the necessary views of the most complicated and crooked 
structure. 



MECHANICAL DRAWING. 



17 




18 INTERMEDIATE COURSE. 

Fig. 60. — To draw a model of a corner closet, 3" high, with two sides perpendic- 
ular to each other, and If n wide, and a third side forming the hypothenuse of the right- 
angled triangle and having a central opening 1J" high and V wide, all the sides and 
ends beings \" thick. 

This Figure is designed entirely for an exercise in projection. It is really an 
example of bad drawing, because more than half of the views are superfluous, and 
one of the requisites of a good drawing is that it should contain no views but what are 
necessary to give useful information. 

Draw the plan, A, with one of its square sides perpendicular to the front plane, 
and from it project an elevation, B, on a plane parallel with the inclined side. Pro- 
ject a front view, C, and a side view, D, as shown, making the latter a section on 
the vertical plane ab. From the plan project a central vertical section, E, on a plane 
perpendicular to the inclined side, and from E project an oblique rear view, F, on a 
plane perpendicular to the plane of projection of E. From the elevation, B, project 
an oblique side view, G, on a plane perpendicular to that of B. 



Each Figure should be inked as soon as completed, and according to the direc- 
tions already given — that is, the fine black lines and dotted lines first, and then the 
heavy black lines, which will complete the lines of the object. Then ink the blue 



MECHANICAL DRAWING. 19 

centre lines, ab, cd, c f d f , c"d" , ef e'f, and gh. Then ink the red dimension lines and 
put in the dimensions, points and letters in black. 

Using Plate 9 as a guide, the student should devise other solids and draw views 
of them from various points, in order to become familiar and expert in treating oblique 
views and in obtaining projections parallel to any desired part. 



DEVELOPMENTS— HEXAGONAL PRISM AND 

CYLINDER. 

PLATE io. 

Fig. 61. — To draw the Development of the entire surface of a Hexagonal Prism, 
2" diameter and 2 J" high, cut off at an angle of 45° as shown. 

First draw the projections of the Prism, and, if any difficulty is experienced, refer 
to Plate 7, Fig. 52, of the Junior Course, which is similar. 

By the Development of the surface is meant unfolding it into one plane. All 
the sides and ends of this prism are plane surfaces, but each is at an angle with 
the others, and the requirement is to draw them all in one plane in their proper 



20 INTERMEDIATE COURSE. 

relative positions so that their outline could be cut out from the paper, or other 
material on which they were drawn, and the surfaces then be folded up to form 
the prism. 

The top- view shows the true width of the six vertical sides of the hexagon, 
therefore, take this width in the spacing dividers and step it off six times on a 
horizontal line, and from each of these points erect a perpendicular to this line. 
Imagine the surface cut on the line ab, the intersection of the two largest sides, 
and unfolded from that line, then ab will be the length of the extreme vertical 
lines a!b ! of the development. The line cd will be the length of the intersections 
of the front and rear sides with these largest sides ; and if we make c'd' and c"d" 
equal to o,d and draw b'd' and b ff d n ', we will have these largest sides represented in 
their true size. In like manner, lay off the length ef on the next vertical lines of the 
development as e'f and e"f f , and gh on the central one as g'h* ', and connect d'f, fh', 
h/f, and f'd", to complete the development of the sides. Draw the hexagonal base, A y 
exactly as shown in the Plan, and copy the inclined top, B, from the auxiliary view. 
Every face of the solid is now drawn in its true size and in such relation to the 
adjoining faces that, if the figure be cut out from the paper and folded on the remain- 
ing lines, it would produce the exact form required. 

This development should be repeated on cardboard, the outlines cut out, 
the other intersections cut partially to facilitate folding and the object be 
produced. 



MECHANICAL DRAWING. 21 

Fig. 62. To draiv the projections and development of a Cylinder, whose base is 
perpendicular to the axis and whose top is inclined. 

Let the Cylinder be 2" diameter and the top be inclined at an angle of 45° with 
the axis, and let the extreme height of the Cylinder be 2-J". 

A Cylinder has a curved surface generated by moving a right line so as to touch 
a curve, all the positions of this right line being parallel. The right line is called 
the generatrix and the curve the directrix. The latter may be a curve of any form, 
but it is only necessary to consider the case of cylinders generated by moving a right 
line in contact with a circle, all the positions of the line being parallel to each other 
and perpendicular to the plane of the circle. If another right line be passed through 
the centre of the circle perpendicular to its plane it will be the axis of the cylinder, 
and will be parallel to and equidistant from the generatrix in all its positions. The 
surface of the cylinder may thus be considered as made up of an infinite number of 
right lines equidistant from and parallel to the axis. Again, the cylinder may be 
generated by moving the circle, as a generatrix, along the right line as a directrix, 
always keeping the circle parallel with its first position, in which case the surface of 
the cylinder maybe considered as made up of an infinite number of circles, the planes 
of which are perpendicular to the axis. Therefore, any plane perpendicular to the 
axis will cut the cylinder in a circle, and any plane parallel to the axis will cut it 



22 INTERMEDIATE COURSE. 

right Hues. Any point on the surface of a cylinder can be definitely determined by 
passing through the poiut two planes, one perpendicular and the other parallel to the 
axis, for it will be at the intersection of the circle produced by the first plane with the 
right line produced by the second plane. A knowledge of these simple principles will 
enable any required projections of a cylinder to be drawn. 

In the problem under consideration, the cylinder is 2" diameter; hence, describe 
a 1" circle for the top- view, through the centre of which draw a vertical line ab and 
a horizontal line cd for the traces of two vertical planes, one. perpendicular and the 
other parallel to the front vertical plane of projection, the intersection of which will 
be the axis of the cylinder. Draw a horizontal line in the front-view for the base 
of the cylinder and perpendicular projecting lines tangent to the circle in the plan to 
intersect this base. On one of these projecting lines lay off the given height, 2J", 
and through this point, at the given angle, 45°, draw the line of the top to inter- 
sect the other projecting line. This will complete the front-view. Those por- 
tions of the projecting lines included between the horizontal base and the inclined 
top are the only visible lines produced by the curved surface alone. They are 
the lines in which the cylinder would be touched by planes tangent to it and per- 
pendicular to the plane of projection. Hence, the projection of a cylindrical surface 
consists of the traces of planes perpendicular to the planes of projection and tangent 
to the surface. 



MECHANICAL DRAWING. 23 

To proceed with the side-view, draw the vertical line c'd r for the trace of the 
central vertical plane cd and across it project the base, on which lay off the diameter, 
2". At the extremities of this diameter draw indefinite vertical lines for the projec- 
tion of the curved surface. 

As the top is inclined, its projection in the side-view will be a curve, and, in 
order to draw this curve, it will be necessary to find a sufficient number of points 
through which it passes by first determining these points in the top and front-views 
and then locating them in the side-view. To do this, divide the circle in the top-view 
into any number of equal parts and through each of these points draw vertical pro- 
jecting lines to the front- view and also horizontal projecting lines. These projecting 
lines are the traces of two sets of vertical planes intersecting each other in the surface 
of the cylinder. Draw them also on the side-view by laying off from c'd' their 
distances from cd. Then, from each point where a plane intersects the projection 
of the inclined top in the front-view, draw a horizontal projecting line to the side-view 
and where this intersects the corresponding plane in that view will be the correspond- 
ing point of the curve of the top. Through all the points thus found draw a curve 
for the side-view of the top, which, in this case, is a circle. This will complete the 
side-view. 

In order to find the true shape and size of the top it is necessary to project it upon 
an auxiliary plane parallel with it. For this purpose draw, parallel with the front- 
view of the top, the trace c"d" of the central vertical plane cd, and also the traces' 



24 INTERMEDIATE COURSE. 

of the other vertical planes at the same distance from c"d" as they are from cd. Then 
draw projecting lines from all the points in the front-view, which have already been 
determined, to this auxiliary-view ; and, where these lines intersect the corresponding 
vertical planes will be the positions of the points in this view. 

The similarity between this Figure and the hexagonal prism in Fig. 61 should 
be noted, because the curved surface of the cylinder has been divided by lines into a 
number of spaces, and, as regards the lengths of these lines, the treatment is the 
same as if the surface was made up of that number of plane faces. These lines on 
the surface of the cylinder need not be considered as the traces of planes inter- 
secting it, but may be merely as lines at regular distances apart around the circum- 
ference. Then, if these lines are properly drawn in all the views, and a point be 
determined on one of the lines in one of the views, the same point can be readily 
found in all the other views. 

To draw the development of the entire surface of this cylinder, proceed as with 
hexagonal prism in Fig. 61. Draw a horizontal line, and on it lay oif the length of 
the circumference of the cylinder as found by calculation. Divide this length into 
the same number of equal parts as those in the circle in the top-view. At each of 
these divisions erect a perpendicular, and on each perpendicular lay oif the height 
as obtained from the corresponding line in the front-view. A curve passed through 
these points will complete the development of the cylindrical surface. The base will 
be a 2" circle the same as in the top- view, and the top will be an ellipse the 
same as in the auxiliary view. 



MECHANICAL DRAWING. 25 

PYRAMIDS. 

PLATE n. 

Fig. 63. To dravj the development of a 2" Hexagonal Pyramid, 2f " high. 

Draw the Pyramid as in Plate 7, Fig. 51, of the Junior Course. 

Each side of this Pyramid is a triangle, the apex of which coincides with 
the apex of the Pyramid ; hence, select a convenient point for a centre, and, 
with radius equal to the length of a side of the triangle, describe a portion of a 
circle, and on this step off chords the length of the base of the triangle as many 
times as there are triangular sides to the Pyramid, making sure to obtain the 
true length of both the side and the base of the triangle. Connect each of these 
points by right lines with the centre of the circle and with each other for the 
development of the inclined surfaces, and draw the hexagon of the base adjacent to 
the base of either of the triangles. 



Fig. 64. To draw the development of a 2" Hexagonal Pyramid 2J" high, cut of 
by a plane parallel to and § " from the axis, the plane being perpendicular to one of 
the sides of the base. 

First draw the projections of the entire Pyramid and its complete develop- 
ment, as in Fig. 63. Then cut it by the given plane and draw the projections 



26 INTERMEDIATE COURSE. 

of the cut. From the front-view take the length of the line ca, which is left by 
the cat, and lay this length off on the corresponding line in the development. 
The line cb is not shown in its true length in the front-view, because it is not 
parallel with the plane of projection ; but by drawing a horizontal line from the 
point 6 to a line of the Pyramid which is parallel to the plane of projection, the 
true length cb' will be obtained. Cut off the development of the base the same as 
in the plan, and transfer the vertical cut surface from the side-view to the development, 
being careful to bring either of its edges in contact with the proper edge of the rest 
of the development. 

As an exercise, other Prisms and Pyramids, cut in different places, should be 
assumed, their projections drawn, and their developments cut out of cardboard and 
folded up to produce the forms. 

THE CONE. 

PLATE 12. 

Fig. 65. To draw the projections of a Cone 2J" high and 2J" diameter at the 
base, cut off by a plane parallel with and § " from the axis, and to draw the development 
of the entire surface. 

A Cone has a curved surface generated by moving a right line so as to 
touch a curve and at the same time to pass through a fixed point not in the plane 



MECHANICAL DRAWING. 27 

of the curve. The right line is the generatrix, the curve the directrix. Those 
cones only which have a circular directrix, and in which the generatrix passes 
through a point in a line perpendicular to the centre of the plane of the directrix, will 
be considered. This perpendicular line is the axis, the point the apex, and the cone a 
right cone. 

It is evident that no right line can be drawn on the conical surface without 
passing through the apex, and that any plane which passes through the apex will 
cut the surface in right lines, if at all ; also that any plane perpendicular to the 
axis will cut the surface in a circle the radflft of which will be the perpendicular 
distance of any point of the circle from the axis. Hence, if it is required to 
determine any point on the surface, it is only necessary to pass two planes 
through this point, one being perpendicular to the axis and the other containing the 
apex. 

The Cone under consideration has a base 2\" diameter. Locate the centre 
of the base in the top-view and through this centre draw the vertical centre-line 
ab and the horizontal centre-line cd, the former of which will be the trace of a 
vertical plane perpendicular to the front plane of projection and the latter the 
trace of a vertical plane parallel to the front plane of projection. The intersection 
of the planes ab and cd will be the axis of the cone. About this axis, describe a 
circle, 1\ n diameter. Continue ab as a'b' ', which will be the trace of the same 
vertical plane in the front-view. On a'b' lay off the height of the cone, 2J", 



28 INTERMEDIATE COURSE. 

and through the lower point thus marked off draw a horizontal line for the base. 
Project the extremities of the diameter of the base from the top-view to this line, and 
draw inclined lines from these points to intersect in the apex already laid off on the 
axis. * At a convenient distance from a'b' draw the vertical line c'd', which will be 
the trace of the plane cd on the side vertical plane of projection. From the front- 
view project the apex and the line of the base to the side-view and on this line lay off 
the diameter of the base by marking off the radius on each side of the plane c'd' 
which is an elevation of the plane, cd. Then draw the inclined lines from the 
extremities of the base to the apex, thUPsame as in the front-view. 

Having now three views of the entire Cone, it is necessary to draw the plane 
which is to cut it, parallel with the axis and § " from it, according to the proposi- 
tion. Let this cutting-plane be perpendicular to the front. Its trace will be a 
line parallel with and § " from the axis in both top and front-views, but its actual line 
of intersection with the conical surface will be curved and not straight. The true 
shape of this curve of intersection will appear in the side-view, and must be deter- 
mined from the straight lines which constitute its projections in the top and front- 
views. Fix upon any number of points at any distances apart on the line in the 
front-view and through them pass horizontal planes. The projections in the top-view 
of the intersections of these planes with the conical surface will be circles, the radius 
of each of which will be equal to the distance from the axis to the point where the 
plane cuts the conical surface, as obtained from the front- view. In the top-view, 



MECHANICAL DRAWING. 29 

describe these circles so as to cut the vertical cutting-plane. Draw the traces of 
these same horizontal planes in the side-view and on them lay off from the central 
vertical plane c f d f the distances from the same plane cd in the top-view of the points 
where the circular traces of the corresponding horizontal planes intersect the cutting- 
plane. A curve passed through the points thus found will be the true shape of 
the line of intersection of the vertical cutting-plane with the conical surface. This 
curve is a hyperbola. 

The problem thus far could also have been solved by drawing, in the three 
views, the traces upon the conical surface of a series of vertical planes passing through 
the axis, and projecting the points of their intersection with the cutting-plane from 
the front-view to the corresponding traces in the side-view ; but this method is not as 
accurate in this instance on account of the acute angle at which the traces intersect 
the plane. These traces will assist materially, however, in drawing the development 
of the surface of the Cone. 

To construct this development, divide the circle of the base in the top-view into 
any number of equal parts, and draw diameters through these points. These 
diameters will be the traces of a series of vertical planes passing through the axis 
of the Cone. Project the extremities of these diameters to the front and side-views 
of the base, and draw lines from these points on the base to the apex. These lines 
will be the traces upon the conical surface of the same series of vertical planes, and 



30 INTERMEDIATE COURSE. 

as the Cone is symmetrical about the axis, each line will be of the same length and 
will have the same inclination to the axis as the side of the cone. Hence, describe an 
arc of a circle, with radius equal to the length of the side, and step off on this arc 
divisions of the same number and length of arc as those already made in the circle 
of the top-view, and then connect these divisions with the centre, and the result will 
be the development of the entire conical surface. 

To cut away the same portion of the development as has already been done 
of the elevations, it is only necessary to determine in the former the locations of the 
points which have already been fixed in the latter. These points are the intersections 
of the traces of a series of horizontal planes with the vertical cutting plane. Draw 
the traces of the horizontal planes in the development by describing arcs of radii 
equal to the distances from the apex to the traces on the conical surface, as shown in 
the front- view. In order to locate the required points on these arcs, determine their 
positions in relation to the central plane cd in the top-view by drawing lines through 
them from the apex to the circle of the base. Draw these same lines in the develop- 
ment by transferring their points from the circle in the top- view to the arc in the develop- 
ment. Where the lines intersect the corresponding arcs in the development will be the 
points required, and curves drawn through the points will cut away the development to 
correspond with the elevations. Copy the base of the cone from the top-view, and, 
adjoining it, copy the hyperbola from the side-view to complete the development, 



MECHANICAL DRAWING. 31 

which, if cut out from the paper and properly rolled and folded, will produce the 
solid shown in the drawing. 



Fig. 66. To draw the projections of a Cone 2}" high and 2J" diameter at the 
base, cut off by a plane which makes an angle of 60° with the axis, and ivhich cuts the 
surface at a perpendicular distance of 1 inch below the apex, and to draw the develop- 
ment of the entire surface. 

Draw the three views of the entire cone, as in Fig. 65, and, in the front-view, 
draw the trace of the cutting plane as given. Draw an auxiliary view on a plane 
parallel with the cutting plane. 

It is first necessary to determine points on the line of intersection of the cutting 
plane with the conical surface in the front-view, which can be doue by intersecting it 
with a series of either horizontal or vertical planes, whichever can be most readily 
used or produce the most accurate results. It is evident that horizontal planes will 
intersect it at much more acute angles than vertical planes, and that the intersections 
will be less distinctly defined ; hence it is best to use vertical planes, which, as has 
already been shown, must all pass through the apex of the cone. Hence, in the top- 
view, divide the circle of the base into any number of equal parts and draw diameters 
through these divisions for the traces of a series of vertical planes passing through the 
apex. Draw the traces of these planes upon the cone in the three other views. From 



32 INTERMEDIATE COURSE. 

the points in the front-view where these traces intersect the trace of the cutting plane, 
draw projecting lines to intersect the traces in the other views. Curves drawn through 
the latter intersections will give the projections of the cut surface, the true size and 
shape of which will be given in the auxiliary-view, which is a projection on a plane 
parallel with this surface. The shape is an Ellipse, as is always the case when the 
cutting plane passes through both sides of the cone. 

The surface is developed by describing an arc of a circle of radius equal to the 
length of the side of the cone, stepping off on this arc the same divisions as were used 
in the top-view, connecting these divisions with the centre by lines, in order to repre- 
sent upon a plane the traces already used on the conical surface, and laying off the true 
length of these traces, as explained in Fig. 64. A circle 2J" diameter and an ellipse 
copied from the one in the auxiliary view, completes the development, which should be 
cut out of the paper and put into the form of the solid. 



The student should draw several different Cones of different relative heights, and 
should cut them by vertical planes at different distances from the axis, and also by 
inclined planes at different angles with the axis. 



MECHANICAL DRAWING. 33 



PLATE 13. 



Fig. 67. To draw the projections of a Cone 2J" high and 2\ n diameter at the 
base, cut off by a plane parallel to and §" from the side. 

Draw the three views of the entire cone as before, and in the front-view, draw 
a line parallel to the side of the cone and f " from it for the trace of the cutting 
plane. In this instance it will be best to use horizontal planes to intersect the cutting 
plane for the purpose of determining certain points of its intersection with the conical 
surface. Hence, draw the traces of any number of horizontal planes in the front and 
side-views, and, in the top- view, draw the circular traces which these planes make 
upon the surface of the cone. Each of these horizontal planes will draw a trace 
upon the cut surface, and the length of each trace is the chord of the arc of the 
circular trace of the same horizontal plane as shown in the top- view. Hence, from 
the front- view draw projecting lines from the points of intersection of each horizontal 
plane with the cutting plane to intersect the circular traces in the top-view, and these 
intersections will determine the points in relation to the central plane cd. On the 
horizontal traces in the side-view lay off from c'd' on each trace the distances from 
cd in the top-view of the points in the corresponding trace. Curves drawn through 
these points will complete the top and side-views. 



34 INTERMEDIATE COURSE. 

To obtain the true size and shape of the cut surface, the cone should be projected 
upon a plane parallel with this surface. To do this, draw c n d" parallel to the trace 
of the cutting plane in the front-view, for the trace in an auxiliary-view of the cen- 
tral plane cd in the top -view, and then draw the projections of the base and apex of 
the cone in this auxiliary-view, the former of which will be an ellipse, and the latter 
a point. From the apex draw tangents to the ellipse to complete this view of the 
entire cone. Then project upon this view the traces of the horizontal planes upon the 
cut surface, and lay off the lengths of these traces as obtained from either the top or 
side-view. A curve drawn through these points will give the exact shape and size 
of the cut curface. 

This curve is a Parabola, as are all curves formed by the intersection of a right 
cone of any proportions with a plane. parallel to its side. 



Fig. 68. To draw the projections of a Tube 2 J" outside and If" inside diameter, 
cut off by a plane at an angle of 60° with the side, from a point 2J r/ from one end. 

This problem is similar to the one in Fig. 62, but is to be solved by a more 
rapid method, which, although not as accurate, is still sufficiently so for most prac- 
tical purposes. 

Draw the trace ab of the central vertical plane, and in a location convenient 
for the top-view, draw the trace cd of the other central vertical plane, perpen- 
dicular to the first. The intersection of these planes will be the trace of the axis 



MECHANICAL DRAWING. 35 

of the Tube in the top-view. About this point as a centre, describe a circle, 2\" 
diameter, for the exterior of the Tube, and one, If" diameter, for the interior. 
Draw a horizontal line for the front- view of the lower end, set the triangle tangent 
to the circles in the top-view, and draw indefinite lines up from this lower end for 
the front elevation of the Tube, the top being as yet undetermined. Lay off the 
given length 2J", and through this point draw the trace of the cutting plane at the 
given angle, 60°. 

Draw, in the side-view, the trace c'd' of the central vertical plane cd and the base 
and sides of the Tube, leaving the top indefinite. As the line of intersection of a 
cylinder by a plane, which cuts the axis, is either a circle or an ellipse, the side-view 
of the top in this case will be two ellipses, and it is frequently allowable to approxi- 
mate these ellipses by means of circular arcs, a method of finding the centres of which 
is given in Fig. 32, Plate 3, of the Junior Course. 

From the point of intersection of the cutting plane and the axis in the front-view, 
draw a horizontal projecting line across the side-view of the Tube. This will con- 
tain the major axes of the ellipses. Also, from the front-view, project the points where 
the trace of the cutting plane intersects the exterior and interior of the Tube, to the 
line e'd' in the side-view. These points will determine the minor axes. On these 
axes construct approximate ellipses with circular arcs. 

To obtain an auxiliary-view parallel with the cutting plane, draw, parallel with 
this plane, the trace c"d" of the central vertical plane cd, and upon this line project 



36 INTERMEDIATE COURSE. 

from the front-view the external and internal extremities and the centres of the top 
and base. These centre lines will limit the parallel lines of the tube in this view ; 
therefore lay off on one of them the diameters of the tube and draw the lines parallel 
with c"d f '. All the points necessary for drawing approximate ellipses for the top 
and base in this view are now obtained. 



Fig. 69. To intersect a cone of two nappes by a plane forming an angle with the 
axis and cutting both nappes. 

By a cone of two nappes is meant one which is produced by a generatrix which 
continues beyond the apex, the result being two equal and similiar cones having the 
same axis and the same apex, but tapering in opposite directions. 

Let each cone be 4J" high and 3" diameter at the base. Draw the top and 
front-views as shown. Draw the trace of the cutting plane in the front-view so 
that it intersects the lower base If" from one extremity and the upper base \" from 
the opposite extremity. In the top-view draw the lines of intersection of the cutting 
plane with the upper and lower bases. Draw the traces of a series of horizontal 
planes in the front and top-views. Project the points of intersection of the cutting 
plane with these traces in the front-view to the corresponding traces in the top- 
view, and draw the curves through the points thus obtained in the top-view. These 



MECHANICAL DRAWING. 37 

two views give all necessary information, excepting the true form of the cut 
surfaces. These will be Hyperbolas, as is always the case when a cone is inter- 
sected by a plane which does not pass through both sides and is not parallel with 
either. 

To find the form of these surfaces, a projection must be made on a plane parallel 
with the cutting plane by making the trace o!d! of the vertical central plane cd 
parallel with this cutting plane, and about c f d', which, of course, contains the axis, 
completing the auxiliary-view of the entire cone. Then upon this view project from 
the front-view the traces of the intersections of the horizontal planes with the cutting 
plane (the length of these traces being obtained from the top-view), and draw the re- 
sulting hyperbolas. 

A curious fact will then become apparent, namely, that the hyperbola pro- 
duced upon the upper nappe is precisely the same curve in form and size as the 
one on the lower nappe, and this will always occur when the same plane cuts both 
nappes. 



38 INTERMEDIATE COURSE. 

INTERSECTIONS OF SOLIDS HAVING PLANE 

SURFACES. 



PLATE 14 

Fig. 70. To draw a vertical prism \\" square and 3" long, intersected by a hori- 
zontal prism, \\" square and 4" long, one side of each prism to make an angle of 45° 
with the front plane of projection and the axes to intersect at their centres. 

The importance of training the mind so as to be capable of a clear conception 
of the intersections of solids, will be appreciated when the fact is considered that all 
structures contain such intersections, which may be, and often are, the only part 
of the structure where any difficulty is experienced in the designing or drawing. 
The principles involved are really simple, and the solution of apparently complicated 
and difficult cases becomes easy after the habit is acquired of reducing each case into 
its elements. 

To proceed with the present case, locate the traces of the central vertical 
planes ab and cd in the front, top and side-views, and draw the If" square end 



MECHANICAL DRAWING. 39 

of the vertical prism in the top-view. Lay off the height 3" in the front-view, 
and, midway of this height, draw the trace e/ of a central horizontal plane in 
both front and side-views. At the intersection of ef and c'd f draw the \\" 
square end of the horizontal prism, and lay off the length of the latter, 4", in the 
front- view. 

Draw the ends of the vertical prism in the front and side-views and project its 
corners to them from the top- view. Now, as the diagonal of the square of this prism 
is longer than that of the horizontal prism, the front and back corners of the former 
will not touch the latter ; hence the lines of these corners will extend uninterruptedly 
from the top to the bottom ; therefore draw these lines complete in the front and side- 
vie ws. 

The right and left-hand corners of the vertical prism being in the same plane 
with the top and bottom corners of the horizontal prism, these corners will intersect, 
and their points of intersection are immediately obtained from the side-view, as at x, 
the front-view of which is at x f and the top-view at x" . Complete the projections of 
these corners in all the views. 

As the diagonal of the square of the horizontal prism is shorter than that of 
the vertical, the front and back edges of the former prism will intersect the four 
sides of the latter in points as yet undetermined. To find these points, project 
the ends of the horizontal prism from the front to the top-view, lay off the length 



40 INTERMEDIATE COURSE. 

of the diagonal on the latter, draw the corners until they intersect the sides of the ver- 
tical prism, and project these intersections y to the front-view, as y' . 

The points of intersection of all the corners of the horizontal prism are now 
determined in all the views, but, as the sides of both prisms are inclined to the front 
plane of projection, it is evident that the intersection of these sides will show in the 
front- view. These intersections must be right lines, because the sides are planes, 
and the intersections of planes can only be right lines. As the lines of the corners 
are included in the planes of the sides, and as the points of intersections of the corners 
are already determined, the intersections of the sides must be right lines joining these 
points, as x'y'. 



Fig. 71. To draw a vertical prism If" square and 3" long, with its side at an 
angle of 15° with the front plane of projection, intersected by a horizontal prism lj /r 
square and 4" long, with its axis parallel to and its side at an angle of 15° with the front 
plane of projection, the axis of the horizontal prism passing \ n in front of that of the 
vertical prism. 

Draw the traces ah, cd, c'd' ', ef, gh and g f h' of the central vertical and horizontal 
planes which contain the axes of the two prisms, cd being \" in front of gh. About 
the intersection of ah and gh in the top-view, describe the If" square end of the ver- 



MECHANICAL DRAWING. 41 

tical prism, with its side at the given angle of 15°, and about the intersection of ef 
and g r h', in the side-view, draw the 1 \" square end of the horizontal prism Avith its 
side at 15°. 

Complete the projection of the horizontal prism in the top-view and that of the 
vertical prism in the side-view. Then the top-view will immediately give the points 
where the corners of the horizontal prism intersect the sides of the vertical prism, 
and the side-view those where the corners of the vertical intersect the sides of the 
horizontal. Take for instance the front right-hand corner of the vertical and the 
front upper corner of the horizontal prisms. The former intersects at point x in the 
side-view, the other projections of which are x ! in the front-view and x" in the top- 
view. The latter intersects at points y and z in the top-view, y r and z 1 in the front- 
view and y" in the side-view. Lines connecting x' with y' and z' will give the lines 
of intersections of these sides in the front-view. Find the points of intersection of 
the other corners with the other sides in all the views and the lines of intersection of 
the sides in the front-view. 

An analysis of this figure will show that in drawing it the following problems 
have been solved : — the intersection of vertical lines with planes perpendicular to one 
vertical plane of projection, but inclined to the other ; — the intersection of horizontal 
lines with vertical planes inclined to the vertical planes of projection ; — and the inter- 
section of planes which are perpendicular to one but inclined to the other two planes 
of projection. 



42 INTERMEDIATE COURSE. 

Fig. 72. To draw a vertical hexagonal prism 2" diameter and 3J" long with one 
side parallel to the front plane of projection, intersected by an inclined prism lj r/ square 
and 4l" long, the axes intersecting at an angle of Ab° at a, point in the centre of the former 
and If" from the upper end of the latter, the sides of the latter being at 45° with the 
front plane of projection. 

Draw the traces ab, cd and c'd' of the central vertical planes, lay off on ab 
the height, 3J", of the hexagonal prism, draw indefinite lines for its ends, and mark 
the half, If", of this height, through which, at an angle of 45°, draw the trace ef 
of a plane perpendicular to the front plane of projection and containing the axis of the 
inclined square prism. On ef, from its point of intersection with ab, lay off If" 
upwards and 2J" downwards for the length, 4", of the inclined prism and draw 
indefinite lines for its ends. In the top-view, about the intersection of the central ver- 
tical planes ab and cd, describe the 2" hexagonal end of the vertical prism. To draw 
the square end of the inclined prism, select a point on ef through which to pa*ss a 
trace c"d" of the central vertical plane cd upon a plane parallel to the end of the 
inclined prism. Then the plane c n d" must be perpendicular to ef, because ef is per- 
pendicular to the front plaue of projection, to which cd is parallel. About the inter- 
section of c n d" and ef, describe the 1J" square end of the inclined prism with its 
sides at the given angle 45° with c"d n . 

Project the extreme right-hand edge of the vertical prism from the top-view 
to the front-view until it intersects the end of the inclined prism, and then 



MECHANICAL DRAWING. 43 

project this point of intersection to the side-view and the auxiliary-view, drawing 
this much of the edge in these views. Project the lower edge of the square prism 
from the auxiliary-view to the front- view where it will meet the right-hand edge of 
the vertical prism, and will determine their point of intersection, which point is then 
to be projected to the side-view and the edge completed. The point where the lower 
edge of the inclined prism emerges from the base of the vertical prism, is determined 
in the front-view and projected from this to the top-view. The front aud back edges 
of the inclined prism do not touch the vertical prism, but show continuous lines 
in the front, top and side-views, while the upper edge intersects as shown in the 
front-view, from which the points are projected to the top and side-views. The upper 
square end of the inclined prism intersects two sides and the top of the vertical prism, 
the former points being obtained from the top-view, and the latter points from the 
front- view. 

The front and back sides of the vertical prism intersect the sides of the inclined 
prism in lines determined by the auxiliary-view, and from which the projections 
are made. 

So far, every point of intersection has been directly obtained from either 
one or the other of the views, and this can always be done with objects whose 
lines are parallel with the central planes. If one branch of the object is inclined, 
it is only necessary to make an auxiliary-view on a plane perpendicular to the 
central plane of the inclined branch in crier to get the points of intersection 



44 INTERMEDIATE COURSE. 

with it. Although this is not geometrically necessary, it is more accurate, and 
the view thus obtained is very useful in making the drawing more clear and 
complete, and is often necessary to enable a correct construction of the object to be 
made. 

In order to show that the points can be obtained without the auxiliary-view, the 
base of the vertical prism in the latter has not been completed ; but its intersection 
with the lower sides of the inclined prism have been determined in the following 
manner : — 

If the plane of the base of the vertical prism were continued, it would intersect 
the lower end of the inclined prism in a line of which the point x is the projection in 
the front-view and the line x'x" in the top-view, and it intersects the edge of the inclined 
prism in a point y in the front-view, of which y' is the projection in the top-view. 
Hence, y r x f x" shows the projection in the top-view of the intersection of the plane of 
the base of the vertical prism with the inclined prism, and the points where these 
lines cut the hexagon are the ones required. 

In farther illustration of this, let it be required to find the intersection of the 
front side of the vertical prism with the side of the inclined prism without the use of 
the auxiliary-view. Continue the plane of the former in the top-view until it inter- 
sects the ends of the latter, project these points of intersections to the front-view and 
draw a line connecting them, then as much of this line as is contained within the 
former will be the front-view of the required intersection. 



4 



MECHANICAL DRAWING. 45 

It is always possible, and sometimes convenient, to find the intersection of inclined 
sides of objects by using only two views ; but as the purpose of drawing is to make 
the construction clear, and not merely to display knowledge of geometry, the former 
should not be sacrificed to the latter. 

In Fig. 72, the top and front-views fully determine, in a geometrical sense, every 
point of the object, but a good mechanic would be sorely puzzled in attempting to 
construct it from these views alone, unless he were told that the inclined prism was to 
be \\" square. The side-view, however, could be dispensed with, as it conveys no 
additional information. 



Fig. 73. To draw a vertical hexagonal prism 2" diameter and 3J r/ long with one 
side parallel to the front plane of projection, intersected on one side only by another 
hexagonal prism 1 J" diameter at an angle of 30° with the horizontal plane, the front 
side of both prisms to be in the same vertical plane, the planes of the axes to intersect at 
a distance of \\" above the base of the vertical prism, and the end of the inclined, prism 
to be at a distance of 2\' f from this intersection. 

Draw the traces of the central vertical planes ab, cd and c'd' , and about these 
construct the top, front and side-views of the vertical prism. In the front- view lay 



', 



46 INTERMEDIATE COURSE. 

off on ab a point \\" above the base, and through this, at an angle of 60° with ab, 
draw ef for the trace of an oblique plane containing the axis of the inclined prism. 
Perpendicular to this, draw d'd" for the trace in an auxiliary-view of the vertical 
plane cd, and complete this view of the vertical prism. 

As the difference of the diameters of the two prisms is J ;/ , and as their 
front sides are in the same vertical plane, the vertical planes of their axes will 
be \" apart. Therefore, draw the traces gh, g'h ! and g"h n in the top, side and 
auxiliary-views of the central vertical plane of the inclined prism. In the 
auxiliary-view the intersection of g n h" and ef will be the axis of the inclined prism, 
about which draw the \\ n hexagon. On ef lay off 2 J" from ab, and at this 
point draw a perpendicular to ef for the plane of the end of the inclined prism in 
the front-view. 

The points of intersection of the edges of the inclined prism with the vertical 
prism can be obtained directly from the top-view. Take, for instance, the bottom 
edge xy. It intersects at point y in the top-view, which is projected down to y' in 
front- view and then across to y" in side-view ; and so with all the other edges. 
The intersections of the edges of the vertical prism with the inclined prism are 
obtained from the auxiliary-view. 

It will be noted that in this Figure the views which are useful for constructive 



MECHANICAL DRAWING 47 



purposes are top, front and auxiliary- views, while tne side-view could be dis- 
pensed with, excepting that in actual practice it is generally useful and often import- 
ant for other purposes. 



PLATE 15. 

Fig. 74. To draw a vertical Pyramid 8" high, having a triangular base 5" long 
on each side, with the rear side at an angle of 15° with the front plane of projection, the 
Pyramid intersecting a horizontal Prism 8" long, having triangular ends A\" long on 
each side, with the rear side 'parallel with the front plane and at a distance of 1J" back 
of the axis of the Pyramid, the horizontal centre line of the Prism to be 3J" above the 
base of the Pyramid. 

Draw the three views of the vertical pyramid, commencing with the plan of 
its base in the top-view. Also draw the three views of the horizontal prism com- 
mencing with its end in the side-view. The points of intersection can then be 
obtained directly from the side-view, projected across to the front-view and up to the 
top- view. 

It has been already stated that two views are all lhat are necessary to determine 
all the points of a solid ; and, although three views are generally more convenient and 
desirable, yet sometimes it becomes difficult and tedious to employ all the views 



48 INTERMEDIATE COURSE. 

necessary to ootain directly all the points of intersection. As this Figure is a good 
example of the intersection of inclined lines with inclined planes, it will be well to 
analyze it for the purpose of understanding how such points of intersection can be 
obtained from the top and front-views only, without the use of planes of projection 
perpendicular to the inclined planes. 

Let it be required to find where the inclined line forming the front edge of the 
vertical pyramid intersects the two inclined faces of the horizontal prism. It is 
evident that, if we intersect these inclined planes by a vertical plane containing this 
inclined line, the vertical plane will draw traces on the inclined planes and that 
these traces will contain the points of intersection, because the line is contained by the 
plane which makes them. 

In the Figure, the trace of this vertical plane in the top-view is ab, which, pro- 
jected to the front-view, gives a'b' and a'b" as the traces on the inclined faces. 
These traces intersect the inclined line of the front edge of the pyramid at x and x' } 
which are the points required. 

In the same manner for the left-hand edge, draw the trace cd of its vertical 
plane in the top* view, and project this to c'd' and c'd" in the front-view to obtain 
the points y and y r . 

For the right-hand edge, draw the trace, no, in the top-view, extending the 
limiting lines of the inclined faces far enough to obtain the intersections, and project 
to n'o' and n'o" in the front-view to obtain the points z and %' , 



MECHANICAL DRAWING. 49 



PLATE 16. 

Fig. 75. To draio the same Pyramid and Prism as in Fig. 74, the Pyramid 
being in a similar 'position, but the Prism being inclined at an angle of 60° with the per- 
pendicular, its central plane crossing that of the Pyramid at a height of 4J r/ , and its 
vertical side being 2J-" back of the axis of the Pyramid. 

In this case it is doubtful which of the two methods explained with Fig. 74 is 
the quickest and most reliable. If it is desired to find the intersections by projection 
only, a complete auxiliary-view, projected upon a plane parallel with the end of a 
prism, will be required. This would be the clearest and most easily understood by a 
mechanic proposing to make the object ; but, for the purpose of gaining a better 
understanding of the descriptive geometry involved in the use of only two views, it 
will be best to use the latter method. 

Draw the top and front-views of the Pyramid complete, and, in the front- 
view draw the centre line of the prism crossing the axis of the pyramid 4 J" 
above its base and at an angle of 60° with the perpendicular. On this centre line 
draw an auxiliary end-view of the Prism., and from this make its top and front 
projections complete. 



50 INTERMEDIATE COURSE. 

The problem now is similar to the one in Fig. 74 — that is, there are the three 
inclined planes of the Pyramid intersecting the two inclined planes of the prism, with 
this difference, that the latter incline not only to the front vertical plane, but also to 
the side vertical plane. 

As each of the planes of the Pyramid is limited by the corners, it is only neces- 
sary to find the points of intersection of these corners with the sides of the prism. 
Take the line of the front corner and imagine a vertical plane which contains it to be 
passed through the prism. Such a plane would draw the traces ab y a!b' and a'b" 
upon the sides of the prism, and the points x and x r , where these traces cut the line 
in the front-view, will necessarily be the noints of intersection required, which are 
then projected to x n and x'" in the top-view. 

A vertical plane containing the right-hand corner will draw the traces no, 
n'o f and n'o" upon the sides of the prism, but it is necessary to imagine these 
sides extended far enough to reach the points n and o. Where these traces 
intersect the edge will ' give the points z, z' in the front, and z", z' n in the top- 
views. The points of intersection y and y 1 of the left-hand edge are found in the 
same manner. 

Having found the points where the inclined lines of the edges of the vertical 
Pyramid pierce the inclined planes of the sides of the Prism, then lines xy, xz, yz, and 
x'y r , x f z f , y'z' will be the lines of intersections of the inclined planes of the Pyramid 
with the inclined planes of the Prism. 



MECHANICAL DRAWING. 51 



PLATE 17. 



Fig. 76. To" draw a vertical Pyramid 6J" high, having a triangular base 5-f" 
long on each side, ivith the rear side at an angle of 15° with the front plane of projection, 
the Pyramid intersecting a Prism inclined at an angle of 60° with the perpendicular, 
the axis of the former piercing the central plane of the latter at a height of 3} /r above 
the base, the Prism being 9" long and having triangular ends 3J" long on each side, 
with the rear side parallel with the front plane of projection and at a distance of 1 \ n 
behind the axis of the Pyramid. 

Draw the top and front- views of the Pyramid and Prism complete. 

In attempting to find, as before, the points where the front edge of the Pyramid 
pierces the sides of the Prism by drawing the traces on the latter of a vertical plane 
containing the former, it will be discovered that these traces do not reach the line, and 
that, therefore, the line passes entirely outside the Prism. This being the case, it fol- 
lows that the front, edge of the Prism must pierce the sides of the Pyramid, and there- 
fore the traces of a plane containing this' front edge must be drawn upon these sides. 
Either view can be selected for this purpose, whichever is more convenient or accurate. 
In the present instance, draw upon the top-view of the sides of the Pyramid the traces 
of a plane, perpendicular to the front plane of .projection and containing the line of 



52 INTERMEDIATE COURSE. 

the fron edge of the prism. These traces, in the top-view, will be a'b' , a'c', which 
intersect the line in the required points x and y in the top-view, from which are pro- 
jected the points x f and y' in the front-view. 

To find where the bottom edge of the prism pierces the back face of the pyramid, 
project the trace ef of its plane on the back face from the top- view to the front- 
view, as e'f , and where e'f crosses the bottom edge will be the point, z, which is pro- 
jected to z' in the top-view. 

It will now be understood that the method of determining the point where an 
inclined line pierces an inclined plane is often a matter of judicious selection or of 
invention, and that the best possible training for such operations is to assume differ- 
ent plane solids in various positions, and find their intersections. Therefore, the 
remainder of the intersections in this Figure are required to be found without further 
explanation. 



PLATE 18. 

Pig. 77. To draw a vertical Pyramid *l\ n high, having a triangular base 4" long 
on each side, with the rear side at an angle of 15° with the front plane of projection, the 
Pyramid intersecting another Pyramid ( d" long, having also a 4 r/ triangular base, the 
axis of the second Pyramid intersecting that of the fiist at an angle of 60°, and at a 



MECHANICAL DRAWING. 53 

point 3f " from the base of the second and 5" from the base of the first, the rear side of 
the second also making an angle of 15° with the front plane of projection. 

This is a case in which there would be no benefit whatever in making more 
than two views, for the reason that no two sides are parallel, and no plane of 
projection could be selected which would be perpendicular to more than one of them. 
It is an excellent example of the penetration of inclined planes by inclined lines 
and of the intersection of inclined planes. 

Draw the top and front-views of the pyramids complete. Then a vertical plane 
containing the front edge of the vertical pyramid will draw the traces ab and ac 
upon the sides of the inclined pyramid, and where these traces intersect this edge in 
the front-view will be the points where it pierces the sides, which points are then pro- 
jected to the top-view. 

A vertical plane containing the left-hand edge of the vertical pyramid will 
draw the trace de upon the upper side of the inclined pyramid, and where de 
intersects this edge in the front-view will be the point where the edge pierces the 
upper side. But if a trace of this same vertical plane be drawn on the low r er side, 
ife will be found not to intersect the edge ; hence, the edge must pierce the rear side, 
the trace on which is dn, which does intersect the edge, giving the point where it 
pierces the rear side. 

A vertical plane containing the right-hand edge of the vertical pyramid will 



54 INTERMEDIATE COURSE. 

draw the traces fg upon the upper and hm upon the lower sides of the inclined 
pyramid and where fg intersects this edge in the front-view will be the point where 
the edge pierces the upper side, and where hm intersects it will be the point on the 
lower side. t 

The points where the three edges of the vertical pyramid pierce the sides of the 
inclined pyramid being now determined, the next step is to connect these points for 
the lines of intersection made by the sides of the two pyramids. The three points 
on the upper side of the inclined pyramid can be readily connected ; also, the points 
where the front and right-hand edges pierce the lower side ; but a difficulty arises 
with the left-hand edge from the fact that it pierces the rear side, and a single 
line of intersection cannot be drawn on the surface of the inclined pyramid which 
will connect a point on the rear side with two points on the lower side. The infer- 
ence from this is, that the lower edge of the inclined pyramid must pierce the ver- 
tical pyramid. 

To find these points, cut the vertical pyramid, in the front-view, by a plane 
perpendicular to the front plane of projection and containing the lower edge of 
the inclined pyramid. This will draw the traces op on the rear and ps on the 
left-hand sides of the vertical pyramid, in the top-view, and where op cuts the 
lower edge, will be the point where it pierces the rear side, and where ps cuts it, 
will be the point where it pierces the left-hand side. Then these points can be 
connected with those previously found to complete the intersection of the two 
pyramids. 



MECHANICAL DRAWING. 55 

The study and analysis of the intersections of Solids with Plane Surfaces 
form such an excellent training for intricate and difficult problems in architectural 
and engineering construction, that the importance of a complete mastery of the subject 
cannot be overestimated. Numerous combinations of solids of various forms should 
be drawn in various positions, and their intersections carefully worked out, when it 
will soon be found that difficulties, which at first seemed insurmountable, can be 
readily overcome. 



INTERSECTIONS OF SOLIDS HAVING CURVED 

SURFACES. 

PLATE 19. 

Fig. 78. To draw a vertical Cylinder 2J" diameter and 2f r/ long, intersected by 
two horizontal cylinders of the dimensions and in the positions shown, and to draw the 
development of all the surfaces. 

As explained in Plate 10, Fig. 62, any plane parallel with the axis of a cylinder 
will cut its surface in straight lines. By using a series of planes to trace lines 
on the surface of a cylinder, and finding the points where these lines pierce the 



56 INTERMEDIATE COURSE. 

intersecting cylinder and then connecting these points, the problem becomes a 
simple one. 

In the present instance, cut the side-views of the cylinders by vertical planes 
(the more the better), draw, in the top-view, the traces of these planes upon the 
surface of the horizontal cylinders, and project the points where these traces pierce 
the vertical cylinder to the traces of the vertical planes in the front-view, for the 
required points of intersection. Then connect these points by straight or curved 
lines as required. 

Take, for instance, the vertical plane ab, in the left-hand side-view. This 
will draw the trace a'a' in the top-view, piercing the vertical cylinder at a/ 9 
which, being projected down to the front-view, will intersect the traces of the 
same plane at the points a"b n , which are two points of the lines of intersection 
of the cylinders. Any number of points can be determined in the same manner, 
and a line connecting them (in this case a straight line) will be the intersection. 
In the right-hand side-view, the vertical plane cd draws the traces cV in the top, 
and c"c", d"d" in the front-views, the points of intersection c" and d" being pro- 
jected from c' '. 

The problem consists, after all, in simply finding the projections of lines of 
different lengths, these lengths being obtained from whichever view gives them truly. 

The lines already on the drawing give all the data necessary for the 
development. 



MECHANICAL DRAWING. 57 

Unroll the cylindrical surfaces into plane surfaces (as in Plate 10, Fig. 62), aud 
draw upon the plane surfaces all the traces that are upon the cylindrical ones and in 
the same positions, and mark the points of intersection upon these traces. Connect 
these points by lines for the development of the intersections. 



Fig. 79. To draw a similar vertical Cylinder intersected by horizontal Cylinders of 
the dimensions and in the positions shown, and to draw the development of the entire surface. 

This is merely a modification of Fig. 78, and will best serve as an exercise and 
training if left to the ingenuity of the student without further explanation. 



PLATE 20. 

Fig. 80. To draw a vertical Cone having a base 3J /r diameter and sides at 60° 
with the base, intersected by a Cylinder lj r/ diameter whose axis is perpendicular to the 
side of the Cone and intersects the axis of the latter J " above the base, the end of the 
Cylinder being 2\" from this intersection, and the cone being truncated by a plane parallel 
to, and 2 J" from, its base; and to draw the development of the entire surface of 
the solid. 

Draw the top, front and side-views of the cone aud cylinder, with the exception 



58 INTERMEDIATE COURSE. 

of their intersection, which is to be determined. Draw a complete auxiliary-view of 
them on a plane perpendicular to the axis of the cylinder. It now remains to deter- 
mine their intersection. To do this, some method must be found in which the cutting 
of both cylinder and cone by the same plane will draw straight lines on the surface 
of each, the intersection of which lines will give a point of the intersection of the sur- 
faces. As has already been explained, any plane which intersects a cone and at the 
same time passes through its apex will draw straight lines upon its surface, and any 
plane which intersects a cylinder and is parallel to its axis will draw straight lines 
upon its surface. Hence, in the present instance, if we cut the cylinder and cone by 
a series of planes under the above conditions, the intersections of the traces of these 
planes on the surface of the cylinder with those on the surface of the cone will be 
points of intersection of these surfaces. 

The auxiliary-view, being a projection upon a plane perpendicular to the axis of 
the cylinder, presents a means of drawing the required cutting planes, because all 
planes perpendicular to it will be parallel to the axis of the cylinder, and the traces 
ol any number of such planes can be made to cut the apex a of the cone. This view 
of the end of the cylinder, having already been divided into equal parts, 1, 2, 3, 4, 
5, 6, to locate upon the other views the traces which were used in obtaining the 
ellipses of its end, a convenient method will be to use these divisions as points through 
which to pass the new cutting planes. Hence, in this auxiliary- view, draw the trace 
of a plane from the apex a through each of these divisions. All of these planes will 



MECHANICAL DRAWING. 59 

intersect in a line perpendicular to this plane of projection. Draw this line, ab, in 
the front- view to intersect the plane of the base of the cone at b and project b to the 
top-view. 

Draw a line cd, tangent to the base of the cone, in the top and auxiliary- views, 
transfer the points where the cutting planes intersect this tangent in the auxiliary- 
view to the tangent in the top-view, and connect these points with b, in the top-view. 
Then be, bd, etc., in the top-view, will be the traces of the cutting planes upon the 
horizontal plane of the base of the cone, and lines drawn from the apex a to the points 
where these traces intersect the circle of the base will be the traces of the cutting 
planes upon the surface of the cone. Project these traces to the front-view, and then 
their intersections with the traces of the cutting planes already drawn upon the sur- 
face of the cylinder in both top and front-views will be points of the line of intersec- 
tion of the surfaces of the cone and cylinder in these views. Project these points from 
the front-view to the corresponding traces on the surface of the cylinder in the side- 
view for the corresponding points in that view. 

This principle is applicable, no matter what may be the proportions or relations 
of the surfaces, and, in general, if the requirement is to find the intersection of any 
two solids, such solids must be cut by planes whose traces upon the surfaces of the 
solids will be lines, the projections of which can be readily drawn in the different views. 
The fact should be remembered that the intersection of the surfaces of any two solids 
is a line, and that, if these solids be cut by planes, such planes will draw lines on the 



60 INTERMEDIATE COURSE. 

surfaces of the solids which will meet at their intersection, if at all. The selection of 
the best location and arrangement of the planes is what most requires the exercise of 
the reasoning powers. 

To find the intersection of two cones, for instance, cutting planes can be used which 
pass through the apices of both cones, in which case the planes will draw straight lines 
on the surfaces of the cones, and the points where the lines meet will be points of the 
intersection. If the axis of the cones are parallel, cutting planes can be used which are 
perpendicular to these axes, in which case the traces upon the surfaces will be circles, 
and the intersections of the circles will be points of the intersection of the surfaces, 
and the projections of circles are almost as readily handled as those of straight lines. 

In the cases of Spheres, Ellipsoids, and surfaces of revolution, cutting planes 
perpendicular to the axis will draw circles on the surface, and no great difficulty 
need be apprehended in treating them if the principles already explained are understood. 



OBLIQUE AXIS OF SYMMETRY. 

Fig. 81. To draw the projections of a solid whose axis of symmetry is inclined to 
both planes of projection. 

Let the solid be a rectangular block 3" long, V wide, and \" thick, and 
let the front projection of its axis of symmetry be inclined upwards at an angle 



MECHANICAL DRAWING. 61 

of 60° with the horizontal plane, and the top projection inclines forwards at an 
angle of 45° with the front vertical plane. Draw the trace of a horizontal plane 
ab, and intersect it by the traces, cd and ef, of two vertical planes perpendicular to 
each other, and make the intersection of these traces, h, h! , and h", the lower end 
of the inclined axis of symmetry. Draw the projections of this axis of sufficient 
length to exceed that of the block, say to g, g' and g". Now, in order to draw the 
block of the given dimensions, it is only necessary to draw its projection on a plane 
parallel to the axis, and as h'g f is a top-view of this axis, a line a'b' parallel with h'g' 
will be the trace of the horizontal plane ab upon a plane parallel with the axis. 
Hence, if g' be projected to g'" at the same vertical distance from a'b' as that of g 
from ab, then g'"h'" will be a parallel view of the axis upon which the side of the 
block can be drawn, and from which an end-view can be projected to determine the 
thickness. Now, from h'", on h'"g'", lay off 3" and through this point draw the 
trace of a horizontal plane Im, which will pass through the centre of the upper end of 
the block, then I'm', at the same height above ab, will be the trace of this same plane 
in the front and side-views. The thickness being laid off on the top-view, the ends 
can be projected from the auxiliary-view, and the top-view thus completed, from which 
the corners can be projected to the front- view, their vertical distances from ab and I'm' 
being the same as from a'b' and Im. The corners are then projected from the front 
to the side-view, their horizontal distances from ef and rs being the same as from 
e'f and rV. 



62 INTERMEDIATE COURSE. 

These traces of vertical and horizontal planes, ab, cd, ef etc., are called, for 
brevity, centre lines, and are of great utility. They form the bases from which every 
point of an object can be definitely located, and are valuable lines of reference. They 
should always be inked blue, and preserved. 



Fig. 82. To draw the same rectangular block as in Fig. 81, with the front 
projection of its axis of symmetry inclining upwards at an angle of 30° with the 
horizontal plane, and the top projection inclining backwards at an angle of 30° with the 
front vertical plane. 

Draw the three views of the axis, and from the top-view project the axis upon 
a vertical plane parallel to it, as in Fig. 81, and upon this plane, and another perpen- 
dicular to it, draw the block in its true size, together with the horizontal and vertical 
planes of reference or centre lines. Locate these centre lines in the three original 
views, and complete the projections required. 

The change in the inclination of the block in this Figure from that in Fig. 81 
gives a good idea of the infinite variety of positions in which an object can be drawn, 
and of the judgment required in the treatment of oblique views. It is evident that, 
no matter how any line may be inclined, a vertical plane will always contain it, and it 



MECHANICAL DRAWING. 63 

can be projected upon a plane of projection parallel to this vertical plane. Having 
thus the projection of the line in its true length upon a vertical plane, this can be 
considered the front-view of a new series of views upon which the object, however 
complicated it may be, can be completely and readily drawn, and from these views, 
those required can be projected. 

As this is a problem of frequent occurrence in practical work, and one which 
makes a good test of a draughtsman's capacity, the student should draw different 
objects with the axes of symmetry leaning in different directions, and should not 
abandon the subject until attaining thorough familiarity with it. 



REVIEW. 

The surface of every structure is composed of plane surfaces or curved surfaces, 
or the two combined. The intersections of these surfaces are either straight or curved 
lines, and the intersections of lines are points ; hence, points, lines, and surfaces in 
various relations and combinations constitute everything that Mechanical Drawing has 
to deal with. 

A point has neither length, breadth, nor thickness ; a line has length only; and 
a surface has length and breadth, but no thickness. A line may be considered as 



64 INTERMEDIATE COURSE. 

made up of points, as elements, arranged according to some law ; and a surface may 
be considered as made up of lines, as elements, each bearing a certain relation to the 
adjacent ones. 

In analyzing any structure, planes of reference can be established in relation to 
it in any positions and in any number that may be desirable, and the traces of these 
planes can be drawn on several planes of projections, and the points and lines of the 
structure can be located on these planes of projection at the proper distances from 
the traces of the planes of reference, and any number of views of the structure thus 
be made. If the surface of the structure does not contain enough actual lines to 
enable it to be definitely determined, any of its elements can be assumed to be lines 
and treated as such. If the lines are curved, the points where they pierce planes of 
reference can be determined and various projections of them be constructed. 

Most structures are composed of modifications and combinations of the simple 
solids which have been investigated in this Course and the Junior Course, and 
the experience and training to be obtained from a full comprehension of them should 
serve as a complete preparation for all the problems in orthographic projection 
likely to occur in practical work as applied to Engineering, Architecture, or any con- 
structive art, with the exception of the Helix, which will be investigated in the 
Senior Course. 



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